If `tanthetatan5theta=1` , then what is the value of `sin2theta` ?

Options

A. a)0
B. b)`(1)/(2)`
C. c)`(1)/(sqrt(2))`
D. d)`(sqrt(3))/(2)`

Answer

To solve the equation \( \tan \theta \tan 5\theta = 1 \) and find the value of \( \sin 2\theta \), we can follow these steps:

### Step 1: Understand the equation
The equation \( \tan \theta \tan 5\theta = 1 \) implies that \( \tan 5\theta = \frac{1}{\tan \theta} \). This can be rewritten using the cotangent function:
\[ \tan 5\theta = \cot \theta \]

### Step 2: Use the identity for tangent
Using the identity \( \tan A = \cot B \) implies that:
\[ 5\theta = 90^\circ - \theta + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
This simplifies to:
\[ 5\theta + \theta = 90^\circ + n \cdot 180^\circ \]
\[ 6\theta = 90^\circ + n \cdot 180^\circ \]
\[ \theta = \frac{90^\circ + n \cdot 180^\circ}{6} \]

### Step 3: Find specific values of \( \theta \)
For \( n = 0 \):
\[ \theta = \frac{90^\circ}{6} = 15^\circ \]
For \( n = 1 \):
\[ \theta = \frac{90^\circ + 180^\circ}{6} = \frac{270^\circ}{6} = 45^\circ \]
For \( n = 2 \):
\[ \theta = \frac{90^\circ + 360^\circ}{6} = \frac{450^\circ}{6} = 75^\circ \]
We can choose \( \theta = 15^\circ \) or \( \theta = 45^\circ \) for further calculations.

### Step 4: Calculate \( \sin 2\theta \)
Using \( \theta = 15^\circ \):
\[ \sin 2\theta = \sin(2 \times 15^\circ) = \sin 30^\circ = \frac{1}{2} \]

Using \( \theta = 45^\circ \):
\[ \sin 2\theta = \sin(2 \times 45^\circ) = \sin 90^\circ = 1 \]

### Conclusion
The possible values of \( \sin 2\theta \) are \( \frac{1}{2} \) and \( 1 \).

### Final Answer
The value of \( \sin 2\theta \) can be \( \frac{1}{2} \) or \( 1 \) depending on the value of \( \theta \).

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